Unless you have some really high quality desktop speakers/headphones, I doubt you'd notice the difference over most onboard audio now days. If you have high quality headphones, its pretty much all of the above. However, you'd notice little difference, if any, with any sub 100 USD desktop speakers/headphones. If you do have a nice set of speakers, then its worth investing into a nice sound card to get the most out of them.

Better components used (DAC, capacitors, DSP) = less interference from other devices, better audio sampling, less background/white/unwanted noise in the signal => better quality audio (you could say more clear, livelier, more distinguishable sounds).

Better (or stronger) AMPs (amplifiers) used = louder sounds, some sounds "added" which you colud not hear before (but they were there) esspecialy if you use high impedance headphones which require a stronger (amplified) audio source.

EAX 4/5 or GX (Eax emulation that works) = additional sound effects.

Better sound orientation in games & more sound options.

I've gone from onboard VIA Vinyl HD VT1708S to a Asus Xonar DG card.
Difference on my Logitech S-220 speakers - minimal (the speakers are low end and most audio I listen to is 128 kbps mp3).
Difference on my Creative Fatal1ty headset - vast/large. Sound has more details. Great for gaming.
And the VIA VT1708S is one of the better integrated audio chips I have tried/used/heard.

If you spend good money on a top of the line sound card, you'll need a high end audio setup to take advantage of it, otherwise just stick to onboard audio as you won't notice the difference. These days onboard audio on modern motherboards will be more than sufficient for most audio configurations.